Jackknife variance estimator (${\hat{V}}_J$)

The jackknife variance estimator estimates the sampling variance of the full-data estimator, using the deletion estimates and their mean from jackknife resampling:The square root is the estimated standard error. The factor $n-1$ multiplies the divisor-$n$ variance of the deletion estimates: those estimates vary less than estimates from new independent experiments because almost all their observations are shared. Ryan Tibshirani's ^[ref] give the same scaling in terms of the divisor-$(n-1)$ sample variance of the deletion estimates.

For the sample mean, this formula has an exact algebraic simplification. With $s_n^2$ denoting the usual sample variance,and thereforeUnder the independent and identically distributed assumption and a finite second moment, $s_n^2$ is an unbiased estimator of the population variance ${\sigma }^2$. Consequently ${\hat{V}}_J$ is an unbiased estimator of the sample mean's sampling variance ${\sigma }^2/n$. The equality ${\hat{V}}_J=s_n^2/n$ itself holds for any observed data with $n>1$; its sampling interpretation uses the stated assumptions.

For $(1,2,3,4,10)$, the sum of squared deviations from the sample mean is $50$, giving sample variance $s_5^2=50/4=12.5$. The deletion sample means have squared deviations totaling $3.125$, soHere $12.5$ estimates the variance of individual observations, while $2.5$ estimates the sampling variance of their sample mean. These answer different questions.

For more general statistics, validity requires regularity beyond the deletion formula, such as a sufficiently smooth function of sample means with suitable moments. Ordinary delete-one jackknife resampling can fail for nonsmooth quantiles, including the median; see Larry Wasserman's ^[ref]. With statistical dependence, deleting individual observations need not reproduce the relevant sampling variance; the deletion scheme must reflect that statistical dependence. Shalizi's ^[ref] distinguish the basic independent and identically distributed case from extensions.

For independent identically distributed observations with finite fourth moment, the sample mean of their squares has sampling variance $\mathrm{Var}(X^2)/n$. Its jackknife variance estimator is exactly the displayed unbiased estimator, since its leave-one-out values differ from their average by $-(x_i^2-\overline{x^2})/(n-1)$. The conditional bootstrap variance of a sample mean is ${\hat{v}}_B=n^{-2}{\sum }_i(x_i^2-\overline{x^2}{)}^2=(n-1){\hat{v}}_J/n$. Thus the two methods have a known finite-sample scaling difference, despite sharing the same large-sample target.